Enumeration modulo 4 of overpartitions wherein only even parts can be overlined
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866910760957902848 |
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| author | Carlson, Aidan Hopkins, Brian Sellers, James A. |
| author_facet | Carlson, Aidan Hopkins, Brian Sellers, James A. |
| contents | In 2014, as part of a larger study of overpartitions with restrictions of the overlined parts based on residue classes, Munagi and Sellers defined $d_2(n)$ as the number of overpartitions of weight $n$ wherein only even parts can be overlined. As part of that work, they used a generating function approach to prove a parity characterization for $d_2(n)$. In this note, we give a combinatorial proof of their result and extend it to a modulus 4 characterization; we provide both generating function and combinatorial proofs of this stronger result. The combinatorial arguments incorporate classical involutions of Franklin, Glaisher, and Sylvester, along with a recent involution of van Leeuwen and methods new with this work. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2411_03077 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Enumeration modulo 4 of overpartitions wherein only even parts can be overlined Carlson, Aidan Hopkins, Brian Sellers, James A. Combinatorics Number Theory 11P83 (Primary) 05A17, 05A19 (Secondary) In 2014, as part of a larger study of overpartitions with restrictions of the overlined parts based on residue classes, Munagi and Sellers defined $d_2(n)$ as the number of overpartitions of weight $n$ wherein only even parts can be overlined. As part of that work, they used a generating function approach to prove a parity characterization for $d_2(n)$. In this note, we give a combinatorial proof of their result and extend it to a modulus 4 characterization; we provide both generating function and combinatorial proofs of this stronger result. The combinatorial arguments incorporate classical involutions of Franklin, Glaisher, and Sylvester, along with a recent involution of van Leeuwen and methods new with this work. |
| title | Enumeration modulo 4 of overpartitions wherein only even parts can be overlined |
| topic | Combinatorics Number Theory 11P83 (Primary) 05A17, 05A19 (Secondary) |
| url | https://arxiv.org/abs/2411.03077 |